{smcl}
{* 29jan2014}{...}
{cmd:help mata mm_integrate()}
{hline}

{title:Title}

{p 4 4 2}
{bf:mm_integrate() -- Numerical evaluation of a definite integral of a one dimensional function}


{title:Syntax}

{p 4 8 2}
Integration using Simpson's rule (quadratic interpolation):

{p 8 12 2}
{it:real scalar}
{cmd:mm_integrate_sr(}{it:f}{cmd:,} {it:a}{cmd:,} {it:b}{cmd:,} {it:n} [{cmd:,} {it:sc}, {it:...}]{cmd:)}

{p 4 8 2}
Integration using Simpson's 3/8 rule  (cubic interpolation):

{p 8 12 2}
{it:real scalar}
{cmd:mm_integrate_38(}{it:f}{cmd:,} {it:a}{cmd:,} {it:b}{cmd:,} {it:n} [{cmd:,} {it:sc}, {it:...}]{cmd:)}

{p 4 8 2}
where

{p 12 16 2}
{it:f}:  {it:pointer scalar} containing address of function to be integrated; type {cmd:&}{it:funcname}{cmd:()}

{p 12 16 2}
{it:a}:  {it:real scalar} containing lower limit of integral

{p 12 16 2}
{it:b}:  {it:real scalar} containing upper limit of integral

{p 12 16 2}
{it:n}:  {it:real scalar} containing number of intervals
{p_end}
{p 16 16 2}
{it:n} must be a multiple of two for {cmd:mm_integrate_sr()}
{p_end}
{p 16 16 2}
{it:n} must and a multiple of three for {cmd:mm_integrate_38()}

{p 11 16 2}
{it:sc}: {it:real scalar} indicating that function {it:f} is scalar

{p 10 16 2}
{it:...}:  up to 10 additional arguments to pass on to function {it:f}


{title:Description}

{p 4 4 2} {cmd:mm_integrate_sr()} evaluates the integral of function {it:f}
over a regular grid from {it:a} to {it:b} using Simpson's rule (quadratic
approximation). The evaluation grid contains {it:n} equally-spaced intervals of
width {it:d} = ({it:b}-{it:a})/{it:n}. Let {it:x}_{it:i} = {it:a} + 
{it:i}*{it:d}, then the integral is computed as

{p 8 8 2}
    {it:d}/3 * [{it:f}({it:x}_0) + 4*{it:f}({it:x}_1) + 2*{it:f}({it:x}_2) 
    + 4*{it:f}({it:x}_3) + 2*{it:f}({it:x}_4) + ... + 
    4*{it:f}({it:x}_({it:n}-1)) + {it:f}({it:x}_{it:n})]

{p 4 4 2} Likewise, {cmd:mm_integrate_38()} evaluates the integral of 
function {it:f} using Simpson's 3/8 rule (cubic approximation), computed as

{p 8 8 2}
    3*{it:d}/8 * [{it:f}({it:x}_0) + 3*{it:f}({it:x}_1) + 3*{it:f}({it:x}_2) 
    + 2*{it:f}({it:x}_3) + 3*{it:f}({it:x}_4) + 3*{it:f}({it:x}_5) + 2*{it:f}({it:x}_6) +... + 
    3*{it:f}({it:x}_({it:n}-2)) + 3*{it:f}({it:x}_({it:n}-1)) + {it:f}({it:x}_{it:n})]

{p 4 4 2}
{cmd:mm_integrate_sr()} and {cmd:mm_integrate_38()} assume {it:f} to accept a
column vector of {it:x} values as first argument so that all {it:f}({it:x})'s
can be computed in one call. Alternatively, if {it:f} is scalar, specify
{it:sc}!=0 to call {it:f} for each {it:x} individually. Note that computing
the {it:f}({it:x})'s in one call is usually much faster than using individual
calls for each {it:f}({it:x}).

{p 4 4 2}
{it:f} cannot be a built-in function. To integrate a built-in function you need
to create your own version of it. An example can be found below. Also 
see "Passing built-in functions" in {bf:{help m2_ftof:[M-2] ftof}}.

{title:Remarks}

{p 4 4 2} Example: Integrating the standard normal density from -1 to 1

        {com}: function myf(x) return(normalden(x))

        : I = mm_integrate_sr(&myf(), -1, 1, 1000)

        : I
          .6826894921

        : reldif(I, normal(1)-normal(-1))
          5.11338e-14

        : I = mm_integrate_38(&myf(), -1, 1, 999)

        : I
          .6826894921

        : reldif(I, normal(1)-normal(-1))
          1.15463e-13{txt}

{p 4 4 2}Of course there is no point in integrating {cmd:normalden()} as
{cmd:normal()} already provides appropriate integrals. Nonetheless, let us
further use {cmd:normalden()} to illustrate passing on arguments to {it:f}:

        {com}: function myf2(x, mean, sd) return(normalden(x, mean, sd))

        : mean = 4; sd = 2.5

        : mm_integrate_sr(&myf2(), 0, 3, 1000, 0, mean, sd)
          .2897789667

        : mm_integrate_38(&myf2(), 0, 3, 999, 0, mean, sd)
          .2897789667{txt}

{p 4 4 2}The fourth argument, {it:sc}=0, in the example above tells
{cmd:mm_integrate_sr()} and {cmd:mm_integrate_38()} that it is ok to pass a
vector of {it:x} values to {it:f} (the default). If the function you want to
integrate does not allow element-by-element computations specify {it:sc}!=0
to call {it:f} individually for each x value. Example:

        {com}: real scalar myf3(real scalar x, real scalar k)
        > {
        >     return(normalden(x) * ((1 - (x/k)^2) * (1 - 5*(x/k)^2)))
        > }

        : k = 1.5

        : mm_integrate_sr(&myf3(), -k, k, 1000, 1, k)
        .1445150517

        : mm_integrate_38(&myf3(), -k, k, 999, 1, k)
        .1445150517{txt}

{p 4 4 2}Note that the above function could easily be revised to perform 
element-by-element computations, as is possible in many cases. Example:

        {com}: real matrix myf4(real matrix x, real scalar k)
        > {
        >     return(normalden(x) :* ((1 :- (x/k):^2) :* (1 :- 5*(x/k):^2)))
        > }

        : k = 1.5

        : mm_integrate_sr(&myf4(), -k, k, 1000, 0, k)
          .1445150517

        : mm_integrate_38(&myf4(), -k, k, 999, 0, k)
          .1445150517{txt}

{p 4 4 2}{cmd:mm_integrate_sr()} versus {cmd:mm_integrate_38()}: The
literature claims that {cmd:mm_integrate_38()} has a smaller bias. However,
as soon as {it:n} is moderately large both algorithms perform equally well.

{p 4 4 2}Size of {it:n}: The larger {it:n}, the better the approximation but
the slower the evaluation. In most situations, about 1000 should be
sufficient. However, how accurate your results need to be depends on your
application.


{title:Conformability}

    {cmd:mm_integrate_sr(}{it:f}{cmd:,} {it:a}{cmd:,} {it:b}{cmd:,} {it:n}{cmd:,} {it:sc}{cmd:,} {it:...}{cmd:)}
           {it:f}: 1 {it:x} 1
           {it:a}: 1 {it:x} 1
           {it:b}: 1 {it:x} 1
           {it:n}: 1 {it:x} 1
          {it:sc}: 1 {it:x} 1
         {it:...}:  (depending on function {it:f})
      {it:result}: 1 {it:x} 1.

    {cmd:mm_integrate_38(}{it:f}{cmd:,} {it:a}{cmd:,} {it:b}{cmd:,} {it:n}{cmd:,} {it:sc}{cmd:,} {it:...}{cmd:)}
           {it:f}: 1 {it:x} 1
           {it:a}: 1 {it:x} 1
           {it:b}: 1 {it:x} 1
           {it:n}: 1 {it:x} 1
          {it:sc}: 1 {it:x} 1
         {it:...}:  (depending on function {it:f})
      {it:result}: 1 {it:x} 1.

{title:Diagnostics}

{p 4 4 2}
None.


{title:Source code}

{p 4 4 2}
{help moremata_source##mm_integrate:mm_integrate.mata}


{title:Author}

{p 4 4 2} Ben Jann, University of Bern, jann@soz.unibe.ch


{title:Also see}

{p 4 13 2}
Online: {bf:{help m2_ftof:[M-2] ftof}},
{bf:{help moremata}}
